INCLUSION PROBABILITY PROPORTIONAL TO SIZE

                                                                                                          -Prakhar Srivastava

 

INTRODUCTION:

Sample Survey Design is an integral aspect of Statistical Analysis. While designing a sample survey, it is also equally important to choose a suitable variance estimation method.

The sampling design is created by constructing a sample space in which the population units occur in proportion to their sizes and every pair of units occur in such a way as to provide a non-negative and stable variance estimator. The selection procedure is then straightforward and the values of the inclusion probabilities of pairs of units can be computed easily.

 

IMPORTANCE OF INCLUSION PROBABILITY:

 

1.     Knowing the inclusion probability of each element in the population is very important for sampling theory. Inclusion probabilities give an insight into how the RSS design has more control over which element enters the sample than SRS has.

2.     In probabilistic sampling, each element of the population must have a non-zero probability of being included in a sample, otherwise unbiased estimation is not possible.

3.     The reason, therefore, is straightforward: If not all elements of the population have a positive probability of becoming part of a sample, one cannot expect that an actual sample can describe the unknown population parameters correctly.

4.     The inclusion probability πi refers to the chance that the ith population element becomes part of a sample.

5.      The inclusion probability should be distinguished from the selection probability p(s) of a sample that is the probability that a certain unordered set of elements (e.g., a number of trees included by a sample plot) is selected as a sample.

 

IMPORTANT FORMULAE FOR INCLUSION PROBABILITY:

 

1.    Horvitz and Thompson


 

2.    Yates Grundy


PROPERTIES OF INCLUSION PROBABILITY:

 



APPLICATION IN R

Computes the first-order inclusion probability of each unit in the population given a fixed sample size design








Thus, this example gives proper inclusion of each variable, and with a proper sum of inclusion as 2.

 

REFERENCES

1.     IPPS Sampling Methods

https://www.researchgate.net/publication/277259216_A_class_of_ipps_sampling_schemes

 

2.     Pik: Inclusion Probabilities for Fixed Size Without Replacement

https://rdrr.io/cran/TeachingSampling/man/Pik.html

 

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